Question

find the quadratic polynomial whose zeroes are -2and -5 . verify the relationship between zeros and coefficients of the polynomial​

Answer 1

Given that , The Quadratic Polynomial whose zeroes are -2 & -5 .

Need To Find : The Quadratic Polynomial & verify the relationship between zeros and coefficients of the polynomial ?

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$\qquad \qquad \underline{\pmb{\mathbb{\bigstar \:\:QUADRATIC \:\:POLYNOMIAL \:\::\:}}}$

• The zeroes of Quadratic polynomial are : -2 & -5 .

$\qquad \underline {\boxed {\pmb{ \:\maltese \:Sum \:\: of \:\:zeroes \:\:\purple{ \:(\: \alpha + \beta \;)\:} \:: \: }}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:+\:\beta \:\Big\} \:\: \\\\\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:+\:\beta \:\Big\}\:\:=\:\:\:\:\Big\{ -2 + (-5)\Big\}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:+\:\beta \:\Big\}\:\:=\:\:\:\:\Big\{ -2 -5\Big\}\\\\\dashrightarrow \sf \: \alpha \:+\:\beta \: \:\:=\:\:-7\\\\\dashrightarrow \sf \: \alpha \:+\:\beta \: \:\:=\:\:-7 \:\\\\\dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: \alpha \:+\:\beta \: \:\:=\:\:-7 \:\:}}}}}\:\:\bigstar$

⠀AND ,

$\qquad \underline {\boxed {\pmb{ \:\maltese \:Product \:\: of \:\:zeroes \:\:\purple{ \:(\: \alpha \beta \;)\:} \:: \: }}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:\:\beta \:\Big\} \:\: \\\\\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:\:\beta \:\Big\}\:\:=\:\:\:\:\Big\{ -2 \times (-5)\Big\}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:\:\beta \:\Big\}\:\:=\:\:\:\:\Big\{ -2 \times -5\Big\}\\\\\dashrightarrow \sf \: \alpha \:\:\beta \: \:\:=\:\:10\\\\\dashrightarrow \sf \: \alpha \:\:\beta \: \:\:=\:\:10 \:\\\\ \dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: \alpha \:\:\beta \: \:\:=\:\:-7 \:\:}}}}}\:\:\bigstar$

As , We know that ,

$\qquad \dag\:\:\bigg\lgroup \pmb{\sf \:Quadratic \:Polynomial\:\::\:x^2 \: - \:\{ \:Sum \:of \:zeroes \:\} x\:+ \:\{ \:Product \:of \:zeroes \} \: }\bigg\rgroup$

⠀⠀Here , Sum of Zeroes are α + β & Product of zeroes are α β

$\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: - \:\{ \:Sum \:of \:zeroes \:\}x \:+ \:\{ \:Product \:of \:zeroes \} \:$

$\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: - \:\{ \:\alpha + \beta \:\}x \:+ \:\{ \:\alpha \beta \} \:$

$\qquad \underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: - \:\{ \:\alpha + \beta \:\}x \:+ \:\{ \:\alpha \beta \} \: \\\\\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: - \:\{ \:-7 \:\} x\:+ \:\{ \:10 \} \: \\\\\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: + \:7x \:+ \:10 \: \\\\\qquad \dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: \:Quadratic \:Polynomial\:\:=\:x^2 \: + \:7x \:+ \:10 \: \:\:}}}}}\:\:\bigstar$

$\qquad \therefore \:\underline{\sf The \:Quadratic \:Polynomial \:is \: \pmb{\bf x^2 \: + \:7x \:+ \:10\:}\:.}$

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$\qquad \underline {\bf \:\: \maltese \:\:\purple { Verification\: \:\:between \:it's \:zeroes \:and \:coefficient's \:\:relationship \:\::\:}}\:$

⠀$\qquad \qquad \underline{\pmb{\mathbb{\bigstar \:\:QUADRATIC \:\:POLYNOMIAL \:\:= \:x^2 \: + \:7x \:+ \:10\:}}}$

$\qquad \underline {\boxed {\pmb{ \:\maltese \:Sum \:\: of \:\:zeroes \:\:\purple{ \:(\: \alpha + \beta \;)\:} \:: \: }}}\\\\\dashrightarrow \sf \bigg( \: \alpha \:+\:\beta \:\bigg) \:\:=\:\:\dfrac{\:-\:(\:Cofficient\:of \:x\:)}{Cofficient \:of \: x^2 \:}\\\\\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\\dashrightarrow \sf \bigg( \: \alpha \:+\:\beta \:\bigg) \:\:=\:\:\dfrac{\:-\:(\:Cofficient\:of \:x\:)}{Cofficient \:of \: x^2 \:}\\\\\dashrightarrow \sf \bigg( \: -7\:\bigg) \:\:=\:\:\dfrac{\:-\:(\:7\:)}{\:1 \:}\\\\\dashrightarrow \sf \: -7 \: \:\:=\:\:\dfrac{\:-\:7\:}{\:1 \:}\\\\\dashrightarrow \sf \: \: -7 \: \:\:=\:\:-7 \:\\\\\dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: -7 \: \:\:=\:\:-7 \:\:}}}}}\:\:\bigstar$

$\qquad \underline {\boxed {\pmb{ \:\maltese \:Product \:\: of \:\:zeroes \:\:\purple{ \:(\: \alpha \beta \;)\:} \:: \: }}}\\\\\dashrightarrow \sf \bigg( \: \alpha \:\:\beta \:\bigg) \:\:=\:\:\dfrac{\:Constant \:Term \:}{Cofficient \:of \: x^2 \:}\\\\\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\\dashrightarrow \sf \bigg( \: \alpha \:\:\beta \:\bigg) \:\:=\:\:\dfrac{\:Constant\:Term \: }{Cofficient \:of \: x^2 \:}\\\\\dashrightarrow \sf \bigg( \: 10\:\bigg) \:\:=\:\:\dfrac{\:10}{\:1 \:}\\\\\dashrightarrow \sf \: 10 \: \:\:=\:\:\dfrac{\:10\:}{\:1 \:}\\\\\dashrightarrow \sf \: 10 \: \:\:=\:\:10 \:\\\\\dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: 10 \: \:\:=\:\:10 \:\:}}}}}\:\:\bigstar$

$\qquad \underline {\pmb{\bf Hence, \:Verified \:!\:}}$

Answer 2

Answer:

u too

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